Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 304 / 3

Under the global U(1) gauge symmetry, the phases in $\bar{\psi }\psi$ cancel and $\alpha$ is constant, soBoth the fermion term and $F_{\mu \nu }{F}^{\mu \nu }$ are therefore invariant. To extract the Noether current, temporarily promote $\alpha$ to a function. The variation of the action isThe Euler-Lagrange equations then imply the current conservation law ${\partial }_{\mu }{j}^{\mu }=0$.

Perform the infinitesimal local change of integration variablesin the Euclidean path integral for $⟨j^{\mu }(x)\psi (x_1)\bar{\psi }(x_2)⟩$. The vector transformation has no quantum anomaly, so its functional measure is invariant. The action variation supplies $-i\int \alpha {\partial }_{\mu }{j}^{\mu }$, while varying the two charged insertions supplies contact terms at $x_1$ and $x_2$ with opposite signs. Since a change of integration variables cannot change the integral, the coefficient of the arbitrary function $\alpha (x)$ vanishes:This Schwinger-Dyson equation is the position-space Ward-Takahashi identity.

Fourier transformation sends ${\partial }_{\mu }$ to $i{q}_{\mu }$. Write the connected current three-point function as the two full Dirac propagators joined to the amputated vertex:The two contact terms remove one propagator at a time. Multiplying the transformed identity by $G^{-1}(p_1)$ on the left and $G^{-1}(p_2)$ on the right yieldsThis is the momentum-space Ward-Takahashi identity for the full vertex and full propagator.

Taking $q\to 0$ in the Ward-Takahashi identity givesup to the displayed Euclidean $i$ conventions. The ultraviolet divergence of the zero-momentum vertex is therefore exactly the derivative of the fermion self-energy divergence. In renormalization-constant notation this isso the vertex and fermion wave-function renormalization are not independent. The remaining charge renormalization is controlled by photon wave-function renormalization; in a common convention $e_0=Z_3^{-1/2}e$.

The derivation used only an exact change of variables, invariance of the action, and invariance of the measure. It therefore holds nonperturbatively whenever the regulator and definition of the theory preserve the vector symmetry. A symmetry-breaking regulator requires symmetry-restoring counterterms; a genuine quantum anomaly would obstruct the identity, but vector QED has no such anomaly.